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%I #17 May 07 2016 07:52:52
%S 1,1,2,4,10,28,86,278,928,3164,10958,38428,136168,486796,1753660,
%T 6359961,23202408,85093552,313548346,1160248084,4309812532,
%U 16064728072,60070599076,225271863550,847042748378,3192758928650,12061704111576,45662648135238,173204482763760,658180582310888,2505341336035650,9551632787000829,36469897605758744,139443687986144472,533869533407865024,2046496258409861740,7854102611559917914
%N G.f. A(x) satisfies: A( A(x)^3 ) = C(x) * A(x)^2, where C(x) = x + C(x)^2, with A(0)=0, A'(0)=1.
%C The radius of convergence of g.f. A(x) is 1/4.
%C Specific value S = A(1/4) = 0.44982760488955294204795759797171897522321034552221... satisfies:
%C (1) S^2 = 2 * A(S^3),
%C (2) S^4 = 8 * A(S^6/8) / (1 - sqrt(1 - 4*S^3)).
%C Limit a(n)/A000108(n-1) appears to be near 0.6564...
%C The numerical value of this limit is 0.6564415409950121... . - _Vaclav Kotesovec_, May 07 2016
%H Paul D. Hanna, <a href="/A272484/b272484.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) A( A(x-x^2)^3 ) = x * A(x-x^2)^2.
%F (2) A(x-x^2) = Series_Reversion( A(x^3)/x^2 ).
%F (3) A( A(x^3)/x^2 - A(x^3)^2/x^4 ) = x.
%F a(n) ~ c * 4^n / n^(3/2), where c = 0.09258936990935582... . - _Vaclav Kotesovec_, May 07 2016
%e G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 28*x^6 + 86*x^7 + 278*x^8 + 928*x^9 + 3164*x^10 + 10958*x^11 + 38428*x^12 +...
%e such that A( A(x)^3 ) = C(x) * A(x)^2, where C(x) = x + C(x)^2.
%e RELATED SERIES.
%e C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 +...+ A000108(n-1)*x^n +...
%e A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 32*x^6 + 92*x^7 + 284*x^8 + 920*x^9 + 3080*x^10 + 10544*x^11 + 36684*x^12 + 129228*x^13 + 459860*x^14 +...
%e A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 25*x^6 + 72*x^7 + 216*x^8 + 680*x^9 + 2226*x^10 + 7506*x^11 + 25858*x^12 + 90498*x^13 + 320580*x^14 + 1146670*x^15 +...
%e A( A(x)^3 ) = x^3 + 3*x^4 + 9*x^5 + 26*x^6 + 78*x^7 + 243*x^8 + 786*x^9 + 2619*x^10 + 8928*x^11 + 30967*x^12 + 108870*x^13 + 386928*x^14 + 1387560*x^15 +...
%e where A( A(x)^3 ) = C(x)*A(x)^2.
%e A(x-x^2) = x - x^4 + 2*x^7 - 4*x^10 + 4*x^13 + 23*x^16 - 212*x^19 + 1148*x^22 - 4906*x^25 + 16904*x^28 - 41046*x^31 + 6730*x^34 + 713246*x^37 - 5703472*x^40 +...
%e where A(x-x^2) = Series_Reversion( A(x^3)/x^2 ).
%e A( A(x-x^2)^3 ) = x^3 - 2*x^6 + 5*x^9 - 12*x^12 + 20*x^15 + 22*x^18 - 438*x^21 + 2780*x^24 - 13124*x^27 + 50092*x^30 - 145875*x^33 + 201848*x^36 +...
%e where A( A(x-x^2)^3 ) = x * A(x-x^2)^2.
%o (PARI) {a(n) = my(A=x, C=x, X=x+x*O(x^n)); for(i=1, n, C = X + C^2; A = (2*A - subst(A, x, A^3)/(C*A) )); polcoeff(A, n)}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A272483.
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 05 2016
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